Pullback of a sheaf of modules
= Pullback of a sheaf of modules
{title2=$f^*\mathcal E$}
For a morphism of <ringed space>[ringed spaces], the pullback of an $\mathcal O_Y$-module is
$$
f^*\mathcal E=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal E.
$$
The <inverse image functor> supplies the sheaf $f^{-1}\mathcal E$; tensoring with the source structure sheaf supplies the module scalar extension.